A Morse Theoretic Proof of Poisson Lie Convexity

Bohr's inequality says that if $f(z) = \sum^{\infty}_{n = 0} a_n z^n$ is a bounded analytic function on the closed unit disc, then $\sum^{\infty}_{n = 0} \lvert a_n\rvert r^n \leq \Vert f\Vert_{\infty}$ for $0 \leq r \leq 1/3$ and that $1/3$ is sharp. In this paper we give an operator-theoretic proof of Bohr's inequality that is based on von Neumann's inequality. Since our proof is operator-theoretic, our methods extend to several complex variables and to non-commutative situations.We obtain Bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel Hilbert spaces on various higher-dimensional domains, for the non-commutative disc algebra ${\mathcal A}_n$, and for the reduced (respectively full) group C*-algebra of the free group on $n$ generators.We also include an application to Banach algebras. We prove that every Banach algebra has an equivalent norm in which it satisfies a non-unital version of von Neumann's inequality.2000 Mathematical Subject Classification: 47A20, 47A56.

Download Full-text

A Group-Theoretic Proof of Wilson's Theorem

American Mathematical Monthly ◽

10.2307/2308892 ◽

1958 ◽

Vol 65 (2) ◽

pp. 120 ◽

Cited By ~ 2

Author(s):

Walter Feit

Keyword(s):

Theoretic Proof ◽

Wilson’S Theorem

Download Full-text

Recent Advances in Ordinal Analysis: Π12 — CA and Related Systems

Bulletin of Symbolic Logic ◽

10.2307/421132 ◽

1995 ◽

Vol 1 (4) ◽

pp. 468-485 ◽

Cited By ~ 28

Author(s):

Michael Rathjen

Keyword(s):

Proof Theory ◽

Formal System ◽

Ordinal Analysis ◽

The Road ◽

Recent Success ◽

Theoretic Proof ◽

Independence Results ◽

On The Road ◽

Order Arithmetic ◽

Made In

§1. Introduction. The purpose of this paper is, in general, to report the state of the art of ordinal analysis and, in particular, the recent success in obtaining an ordinal analysis for the system of -analysis, which is the subsystem of formal second order arithmetic, Z2, with comprehension confined to -formulae. The same techniques can be used to provide ordinal analyses for theories that are reducible to iterated -comprehension, e.g., -comprehension. The details will be laid out in [28]. Ordinal-theoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. Gentzen fostered hopes that with sufficiently large constructive ordinals one could establish the consistency of analysis, i.e., Z2. Considerable progress has been made in proof theory since Gentzen's tragic death on August 4th, 1945, but an ordinal analysis of Z2 is still something to be sought. However, for reasons that cannot be explained here, -comprehension appears to be the main stumbling block on the road to understanding full comprehension, giving hope for an ordinal analysis of Z2 in the foreseeable future. Roughly speaking, ordinally informative proof theory attaches ordinals in a recursive representation system to proofs in a given formal system; transformations on proofs to certain canonical forms are then partially mirrored by operations on the associated ordinals. Among other things, ordinal analysis of a formal system serves to characterize its provably recursive ordinals, functions and functionals and can yield both conservation and combinatorial independence results.

Download Full-text

A character theoretic proof of the homological sylow theorem

Communications in Algebra ◽

10.1080/00927878508823201 ◽

1985 ◽

Vol 13 (4) ◽

pp. 987-989 ◽

Cited By ~ 1

Author(s):

David B. Surowski

Keyword(s):

Theoretic Proof ◽

Sylow Theorem

Download Full-text